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Full text of "The Bernouilli Equation and the Torricelli Apparatus"

The Bernoulli Equation 

Patrick Bruskiewich, M.Sc. 

Department of Physics and Astronomy, 
University of British Columbia 
6224 Agricultural Road, 
Vancouver, B.C., Canada V6T 1Z1 










A 

t » 






4, 



Bernoulli's Principle: where the velocity of an 
incompressible fluid (gas or liquid) is high, the 
pressure is low and where the velocity is low, 
the pressure is high. 



Bernoulli's Principle makes sense: Fluids will 
flow from a region of higher pressure to a region 
of lower pressure and in doing so increase their 
speed as a result. 



If this were not the case => if the pressure were 
higher at a point of higher velocity then the 
fluid would be slowed down at that point. As 
a result this would be a point of lower velocity. 
This is a contradiction! 



? 



Bernouilli developed an equation of state that 
expresses his principle quantitatively. Assume 
the fluid flow is steady and incompressible. 

fe£ " v 




Volume ~V, 



V^tavwe r V* 




f77777^' 



Consider the work to move a small volume V\ 
of fluid at point 1, 



W 1 = F x Ah = P 1 A 1 Al 1 



(1) 



At point 2 the force exerted on the fluid is 
opposite to the motion so then 



W 2 = - F 2 Al 2 = - P 2 A 2 Al 2 



(2) 



3 



Work is also done on the fluid to counter the 
force of gravity. 



The work done to lift the mass of fluid in volume 
Vi from point 1 to point 2 is 



W g = - mg(y 2 - y\) (3) 

So then the net work W net done on the fluid in 
moving it from point 1 to point 2 is and is given 
by 



W net = W 1 + W 2 + Wg 



= P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 -yi) (4) 



According to the work-energy principle, the net 
work on a system is equal to its change in kinetic 
energy. That is 



1 1 

W net = ARE — -mv 2 - -mvi (5) 



So then 



1 9 1 o 

-rnv 2 - -mv 1 = P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 - Vi) 



(6) 



Gather the 1 and 2 terms together we find 



1 2 

P 2 A 2 Al 2 + -mv 2 + mgy 2 

i t~\ 

PlA 1 Ali + - mv l + m 9V\ 

Z 



R 



We can now factor out the volume on each 
side of the equation, remembering that p = y t 
namely 



1 2 
A 2 AZ 2 (P 2 + ~pv 2 + P9V2) 

1 2 

= AiAZi(Pi + -pv 1 + P^yi) 



(7) 



We know from the continuity equation that A 2 AZ 2 
= AiAZi, which is the volume of the fluid ele- 
ment flowing through the system. 



We can therefore cancel these terms off both 
sides of the equation. 



What we are left with is 



1 2 

P 2 + ~pv 2 + P9V2 
Pi + ~pv{ + pgy ± 



This is Bernoulli's equation 



Since points 1 and 2 can be any points along a 
tube of flow, Bernouilli's equation can be writ- 
ten 



1 2 

p 2 + ~P v 2 + P 9 V2 = constant (8) 

a. 



at every point in the fluid where y is the height 
of the centre of the tube above the reference 
level. 



The Torrjcelli Apparatus 



b* %-y t 




v;*o 



%x -r 



r, 






AX 



H 



The top of the apparatus and the spigot are 
open to the atmosphere so that J\ = P 2 = 

*atm ■ 



Using Bernoulli's equation (> 2 « 0) we have 



~P v i + P 9 Vi = P 9 V2 



(9) 



Solving for v\ we find 



,2 _ 



vf = 2g(v2 - vi) (io) 



so that 



vi = \][2g(y 2 - vi)] (11) 

This is known as Torricelli's theorem, which was 
discovered about 100 years before the Bernoulli 
equation was developed. 



Torricelli was a student of Galileo Galilei. 



Experiment using Torricelli's apparatus 



Objective: measure the distance the flow goes 
before hitting the ground (Ax) and relate this 
to the height of the fluid column (ft = y^ — 2/1). 



Data points for Ax and h are collected and a 
best fit power law curve is graphed. Assume 
that Ax and h are related by a power law given 
by 






V 




Ax = a ft] 



n 



(12) 



h 



where n is to be determined. 



By taking the natural logarithm of the data 
points and graphing the relationship we should 
find that the slope of the graph is the power of 
the relationship. That is 



U[$ 



X 




u[W 



A [In Ax] 
A [In ft] 



= n = slope 



(13) 



10 



Theoretical Prediction 



We know that 



Ax 

v = (14) 

At V J 



The distance m is related to the time it takes 
the drop to fall by 2/1 = §#t 2 , so then 



v = Ax 



9 



2j/i 



= \/[2 £ h] (15) 



where h = y 2 — 2/1 ■ This means that in theory 



Ax = 2[yi h]2 (16) 



where 2/1 is the height of the spigot above the 
ground. Theory therefore tells us that the slope 
of the datapoints on the log-log Ax and h graph 
should be equal to i 



1 1